### Description

"The cost of software failure is truly staggering. Well known

individual cases include the Mars Climate Orbiter failure

(£80 million), Ariane Rocket disaster (£350 million), Pentium

Chip Division failure (£300 million), and more recently the heartbleed

bug (est. £400 million). There are many, many more examples. Even worse,

failures such as one in the Patriot Missile System and another

in the Therac-25 radiation system have cost lives. More generally, a

2008 study by the US government estimated that faulty

software costs the US economy £100 billion

annually.

There are many successful approaches to software verification

(testing, model checking etc). One approach is to find mathematical

proofs that guarantees of software correctness. However, the

complexity of modern software means that hand-written mathematical

proofs can be untrustworthy and this has led to a growing desire for

computer-checked proofs of software correctness.

Programming languages and interactive proof systems like Coq, Agda,

NuPRL and Idris have been developed based on a formal system called

Martin Type Theory. In these systems, we can not only write

programs, but we can also express properties of programs using types,

and write programs to express proofs that our programs are correct.

In this way, both large mathematical theorems such as the Four Colour

Theorem, and large software systems such as the CompCert C compiler

have been formally verified. However, in such large projects, the

issue of scalability arises: how can we use these systems to build large

libraries of verified software in an effective way?

This is related to the problem of reusability and modularity: a

component in a software system should be replaceable by another which

behaves the same way even though it may be constructed in a completely

different way. That is, we need an extensional equality which is

computationally well behaved (that is, we want to run programs using

this equality). Finding such an ty is a fundamental and

difficult problem which has remained unresolved for over 40 years.

But now it looks like we might have a solution! Fields medallist

Vladimir Voevodsky has come up with a completely different take on the

problem by thinking of equalities as paths such as those which occur

in one of the most abstract branches of mathematics, namely homotopy

theory, leading to Homotopy Type Theory (HoTT). In HoTT, two objects

are completely interchangeable if they behave the same way. However,

most presentations of HoTT involve axioms which lack computational

justification and, as a result, we do not have programming languages

or verification systems based upon HoTT. The goal of our project is

to fix that, thereby develop the first of a new breed of HoTT-based

programming languages and verification systems, and develop case

studies which demonstrate the power of HoTT to programmers and

those interested in formal verification.

We are an ideal team to undertake this research because i) we have

unique skills and ideas ranging from the foundations of HoTT to the

implementation and deployment of programming language and verification

tools; and ii) the active collaboration of the most important figures

in the area (including Voevodsky) as well as industrial participation

to ensure that we keep in mind our ultimate goal -- usable programming

language and verification tools."

individual cases include the Mars Climate Orbiter failure

(£80 million), Ariane Rocket disaster (£350 million), Pentium

Chip Division failure (£300 million), and more recently the heartbleed

bug (est. £400 million). There are many, many more examples. Even worse,

failures such as one in the Patriot Missile System and another

in the Therac-25 radiation system have cost lives. More generally, a

2008 study by the US government estimated that faulty

software costs the US economy £100 billion

annually.

There are many successful approaches to software verification

(testing, model checking etc). One approach is to find mathematical

proofs that guarantees of software correctness. However, the

complexity of modern software means that hand-written mathematical

proofs can be untrustworthy and this has led to a growing desire for

computer-checked proofs of software correctness.

Programming languages and interactive proof systems like Coq, Agda,

NuPRL and Idris have been developed based on a formal system called

Martin Type Theory. In these systems, we can not only write

programs, but we can also express properties of programs using types,

and write programs to express proofs that our programs are correct.

In this way, both large mathematical theorems such as the Four Colour

Theorem, and large software systems such as the CompCert C compiler

have been formally verified. However, in such large projects, the

issue of scalability arises: how can we use these systems to build large

libraries of verified software in an effective way?

This is related to the problem of reusability and modularity: a

component in a software system should be replaceable by another which

behaves the same way even though it may be constructed in a completely

different way. That is, we need an extensional equality which is

computationally well behaved (that is, we want to run programs using

this equality). Finding such an ty is a fundamental and

difficult problem which has remained unresolved for over 40 years.

But now it looks like we might have a solution! Fields medallist

Vladimir Voevodsky has come up with a completely different take on the

problem by thinking of equalities as paths such as those which occur

in one of the most abstract branches of mathematics, namely homotopy

theory, leading to Homotopy Type Theory (HoTT). In HoTT, two objects

are completely interchangeable if they behave the same way. However,

most presentations of HoTT involve axioms which lack computational

justification and, as a result, we do not have programming languages

or verification systems based upon HoTT. The goal of our project is

to fix that, thereby develop the first of a new breed of HoTT-based

programming languages and verification systems, and develop case

studies which demonstrate the power of HoTT to programmers and

those interested in formal verification.

We are an ideal team to undertake this research because i) we have

unique skills and ideas ranging from the foundations of HoTT to the

implementation and deployment of programming language and verification

tools; and ii) the active collaboration of the most important figures

in the area (including Voevodsky) as well as industrial participation

to ensure that we keep in mind our ultimate goal -- usable programming

language and verification tools."

Status | Finished |
---|---|

Effective start/end date | 1/04/15 → 30/09/19 |

### Funding

- EPSRC (Engineering and Physical Sciences Research Council): £499,631.00

### Fingerprint

Programming theory

Computer programming languages

Costs

Reusability

Model checking

Rockets

Missiles

Disasters

Scalability

Radiation

Testing